Equation 33 · The Atlas That Refuses to Close
What does this equation mean?
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol S^αβ
S^αβ is part of the quantity the equation computes from the expression on the right.
Symbol P_β
P_β is part of the quantity the equation computes from the expression on the right.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
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What the article says around this equation
Fix the slicing convention that anchors the centroid to the body’s own rest frame — the choice, standard in the spinning-body literature, in which the spin tensor about the centroid satisfies = 0 — and ask how moves as u is varied away from the body’s own rest-frame observer to some other inertial observer boosted at velocity relative to it. Costa and Natário’s synthesis of the classical spinning-body literature gives the answer in closed form: the centroid measured by the boosted observer is displaced from the rest-frame centroid by
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